Optimal control of nonlinear systems to given orbits

Abstract

Using optimal control techniques we derive and demonstrate the use of an explicit single-step control method for directing a nonlinear system to a target orbit and keeping it there. We require that control values remain near the uncontrolled settings. The full nonlinearity of the problem in state space variables is retained. The “one-step” of the control is typically a composition of known or learned maps over (a) the time required to learn the state, (b) the time to compute the control and (c) the time to apply the control. No special targeting is required, yet the time to control is quite rapid. Working with the dynamics of a well-studied nonlinear electrical circuit, we show how this method works efficiently and accurately in two situations: when the known circuit equations are used, and when control is performed only on a Poincaré section of the reconstructed phase space. In each case, because the control rule is known analytically, the control strategy is computationally efficient while retaining high accuracy. The target locations on the selected target trajectory at each control stage are determined dynamically by the initial conditions and the system dynamics, and the target trajectory is an approximation to an unstable periodic orbit of the uncontrolled system. A linear stability analysis shows that dissipation in the dynamical system is essential for reaching a controllable state.